A111Q7 S5bM2T PUBLICATIONS "^FERENCE

NBSIR 73-138

Time Series Forecasting of Higliway

Accident Fatalities

Alexander R. Craw

Institute for Applied Technology

Technical Analysis Division

National Bureau of Standards

Washington, D. C. 20234

March 1973

Final Report

QC

100

.U56

73-138

1973

Prepared for National Highway Traffic Safety Administration

Department of Transportation

Washington. D. C. 20590

NBSIR 73-138

TIME SERIES FORECASTING OF HIGHWAY

ACCIDENT FATALITIES

Alexander R. Craw

Institute for Applied Technology

Technical Analysis Division

National Bureau of Standards

Washington, D. C. 20234

March 1973

Final Report

Prepared for National Highway Traffic Safety Administration

Department of Transportation

Washington, D. C. 20590

U. S. DEPARTMENT OF COMMERCE, Frederick B. Dent, Secretary

NATIONAL BUREAU OF STANDARDS, Richard W. Roberts. Director

ACKNOWLEDGEMENT

Particular thanks are due to Mrs. E. Leyendecker,

who ran all of the computer prograins and carried most of

the computational analysis load. Also, to Dr. Julius

Lieblein for many helpful suggestions and for discussions

that provided insight into what had often been obscure.

TABLE OF CONTENTS

Page

SUMMARY iii

1.0 THE PROBLEM 1

2.0 THE DATA AND AUXILIARY INFORMATION AVAILABLE. . . 1

3.0 READILY AVAILABLE COMPUTER PROGRAMS 2

4.0 PLAN OF ANALYSIS 2

5.0 PRODUCTION RUNS 3

6.0 CONCLUSIONS 8

7 . 0 TABLE 1 9

8.0 TABLE 2 10

9 . 0 TABLE 3 11

10.0 TABLE 4 12

11.0 TABLE 5 13

12.0 TABLE 6 14

13.0 TABLE 7 15

14.0 TABLE 8 16

15.0 TABLE 9 17

16.0 APPENDIX A, EXPONENTIAL METHODS FOR

ANALYSIS AND FORECASTING OF ECONOMIC TIMJE

SERIES A-1

17.0 APPENDIX B, AVAILABLE COMPUTER PROGRAMS

AT NBS FOR TIME SERIES ANALYSIS . B-1

18.0 APPENDIX C, DETAILS OF EXPSMOOTHING ROUTINE . , . C-1

SUMMARY

Using twelve years of time series data on highway

fatalities, the methodology currently employed by the

National Highway Traffic Safety Administration (NHTSA)

to forecast the annual (calendar year) total of highway

accident fatalities were compared with those obtained

by several computer routines based on exponential smoothing

techniques and available at the National Bureau of Standards.

The use of unadjusted and seasonally adjusted data was

also examined.

It is found that there is no coercive evidence to

lead to abandoning the present NHTSA methods in favor

of readily available computer routines based on exponential

smoothing methods

.

Of the methods examined in this study, the best

results were obtained with the EXPSMOOTHING routine

using unadjusted fatality data.

1. The Problem : The Technical Analysis Division has

been requested to examine the National Highway Traffic

Safety Administration's CNHTSA) current methodologies for

forecasting the annual total of highway accident fatalities

given the monthly fatalities respectively for the first

three months of the year, the first six months and finally

the first nine months of the year, make comparisons of the

results of their methodologies with those based on readily

available computer programs at the National Bureau of

Standards for time series analysis and forecasting.

2 . The Data and Auxiliary Information Available : Data for each month are available for the twelve (12) year

period 1960 through 1971. Also available are the outputs

of NHTSA's Computer Program TIMSR4 : Time Series Analysis of

Fatalities and Vehicle-mileage Data. This time series

analysis program provides 12-month moving averages, monthly

seasonal indices, seasonally adjusted data for each month

and the ordered differences between the seasonally adjusted

data ana an uncentered twelve month moving average. The

ordered differences are used to determine for any month of

a time series upper and lower bounds for the trend point

such that the seasonally adjusted value will fall within

those limits with a known frequency.

The TIMSR4 Program does not have an automatic extrapolation Cforecasting ahead) fe^ature. The NHTSA Methods of

forecasting that we were asked to review are discussed in

Section 5.

1
3, Readily Available Computer Programs: A constraint

imposed on the study was to use computer programs that were

readily available at the National Bureau of Standards. Thus,

a comprehensive study of methods is not given here.

Several time-sharing interactive systems were immediately

available. These were DIALCOM, Computer Sciences Corporation

Conversational Executive and General Electric Mark II. Of

these time-sharing programs we chose to use the General

Electric EXPSMOOTHING routine, because of the variety

of options available and the existence of an automatic

extrapolation feature.

The Large Scale Systems STAT-PACK for the UNIVAC

1108 lists several routines for time series analysis. These

are UNIVAC system subroutines and are not necessarily

National Bureau of Standards routines, but they are available

for use on the NBS UNIVAC 1108. The most appropriate

subroutine for the problem at hand appeared to be

GEXSMO - Generalized Exponential Smoothing.

Appendix B contains a discussion of the available

computer programs at NBS as well as a discussion of the background and details of the GEXSMO computer program. Appendices

A and C provide the same information for the EXPSMOOTHING

program.

4. Plan of Analysis:

a. Use the General Electric - EXPSMOOTHING routine

and the UNIVAC STAT-PACK GEXSMO routine.

2
b. Use both seasonally adjusted data and seasonal

factors (as determined by the TIMSR4 program)

and unadjusted data in separate runs.

c. Base forecasts on 4, 6, 8, and 11 years of

data plus data for 3, 6, and 9 months of new

data

»

d. Do c. for each of the NHTSA forecasting

algorithms

.

5. Production Runs : We started production runs using

the General Electric Time-share routine EXPSMOOTHING , using

seasonally adjusted data that was determined by NHTSA ' s TIMSR4

computer program. We always chose the most complete output

option, so that the routine always performed a cyclic

analysis: if a cycle was indicated by the routine, the

analysis part of the program took this into account. (We view

this as a not too interesting bonus of the routine.) Horizon

time was always chosen to exceed the lead time.* The statistical

analysis of the output was accomplished by a mixture of

computer and hand calculations. The results of this group

of calculations are contained in Table 1.

The total amount of data available for forecasting

covered 11 years, with a minimum of 3 years of data considered

*In the EXPSMOOTHING Program, the lead time is the

term used to describe the time (in periods) between

the last period of data one has as data as input and

the period one is most interested in estimating correctly.

The forecast v/ill be optimized according to the lead

time that is entered. The forecast horizon indicates

how many periods beyond the last data point that will

be forecasted, but has nothing to do with the optimization.

(See Appendix A)

.
necessary for suitable application of the exponential

smoothing process. Thus, annual "forecasts" could be

made after 3, 6, and 9 months in each of 8 years. Instead

of analyzing this full set of 24* periods, half of these

were chosen, namely the three periods in those years

with 4, 6, 8, and 11 antecedent years of data.

In the headings of Table 1, and subsequent tables,

the following symbols were adopted:

S = the actual va.lue of the time series datum

(in this case, fatalities)

S = an estimate of S

S = seasonally adjusted value of S v f < ^ (below} A=S-S; if f^ > 01 the estimate israbovelthe

actual value

T. = summation over the lead time

E* = summation over the calendar year (Jan. - Dec.)

ZA/ZS = (signed) fractional error over lead time

E*A/Z* = (signed) fractional error over the calendar

year (Jan. - Dec.)

MAD = mean absolute deviation

a = smoothing constant

k = length of cycle; k = 1 indicated data is not cyclic.

One of the interesting features of Table 1 is

that there is no apparent pattern for the cycles with the

age of the data. The same holds true for the a value (smoo thine

constant) and the order of smoothing. This led us to dupli-

*An additional 3 periods were analyzed for noncalendar twelve-month years, raising the potential sample

size to 27 and the number tested to 15.

4
cate these computations by suppressing the cyclic analysis.

The summary of the results of these computer runs is given

in Table 2. here there does appear to be some reasonableness

in the variation of the smoothing constant with data age.

The comparison of the forecasting results (Table 3) shows

a slight advantage to the case of using the cyclic analysis,

but the results are not heavily in favor of the latter.

In the case of 4 years and 3 months of seasonally

adjusted data, the computer routine selected a = 0.175

(order 2) for the optimum smoothing constant. We were

interested in determing whether or not the minimum was sharp

or flat, so we made runs for a =0.170 and 0.180. The order

of smoothing remained at 2 and the minimum was found to be

flat; only a very slight change in forecast error fraction

could be gained by changing a from .175 to .170 (see Table

4). The change is from 0.0121 to 0.0109.

Table 5 presents the results of duplicate runs using

unadjusted data in the EXPSMOOTHING routine. Cyclic analysis

was permitted and each time the routine selected K=12 for

the dominant cycle.

We also mdde matching runs starting with unadjusted data

on the UNIVAC 1108 program GEXSMO, starting with a given

cycle (seasonality) length of 12 and using 2 initial periods

to determine the initial seasonality factors and other

starting values. The GEXSMO routine alv/ays produces a forecast for the time indicated as the length of the cycle

5
starting from the last input datum. In the language of the

EXPSMOOTEING routine, the lead time is 1 and the horizon

time is the length of the cycle. The summary statistics

for these runs are presented in Table 6.

It is interesting to note that in working with unadjusted data and these two computer routines, it is possible

to compare the seasonally adjusted data and seasonality

factors produced by these routines with those determined

by the TIMSR4 program. We have not made these comparisons,

preferring at this stage to look at the end results.

Table 7 presents the detailed calculations for the

current methodologies employed by NHTSA to forecast the

calendar year fatalities. These methods were developed

in accordance with descriptions furnished by Mr. Donald F.

Mela in his letter of November 30, 1972, and have been

labelled 1, 2 and 3 in the same order as described by Mr.

Mela. Higher order methods are possible by taking into

account more years of input data in the averaging process,

but we have limited ourselves to the first three methods.

The notations used in the column headings of Table 7

are

m = current total for specified months of fatalities

o . ^ m ^ = the corresponding total for specified months of

fatalities for i^h previous year.

6
= the total fatalities for the i^h previous year

A= leap year adjustment factor

Yj = the forecast calendar year fatalities by method j

With this notation the NIITSA forecast methods may be written;

Method 1: = (m^/m.i) y ^

.

Method 2: Y2 =\Y-^

Method 3: Y3 = X {,m^/l

^ Y_

.

k k

Method k: Y, = A (mV^ m .) Z Y , k > 1 ^ ° 1 -i 1 "i

Table 8 presents a composite summary of the fractional

errors by each of the methods considered. In this table

and Table 9, SAC = Seasonally adjusted data with cyclic

analysis.

Considering just the NHTSA's methods, we find a gradual

improvement in the forecast with increasing order of the

method. nowever, in certain specific cases Method 2 gives

slightly better results with methods of order greater than

2. Generally, the error decreases with the shorter lead

time

.

Comparing the four exponential smoothing methods , the

sample evidence seems to favor EXPSMOOTHING using unadjusted

data. However, each of the four methods has a mean absolute

deviation of approximately 1 percent, which is virtually

indistinguishable from that of Method 3.

7
Comparing the "best" NHTSA method (Method 3) with

the "best" exponential smoothing routine we find for

the sample of 12 trials out of the possible 24 forecasts

that method 3 was better 5 times and the best exponential

smoothing method was better on 7 occasions. This is

obtained by comparing the entries in columns headed

E*a/Z*S and Unadj C in Table 8. However, the range of the

absolute deviations of the exponential methods is smaller

than that of any of the methods 1, 2 or 3. Thus, it

would appear that there is a small gain in going to

the exponential smoothing methods.

Our experience in using the EXPSMOOTHING routine has

made us wary of using long lead times, in particular L = 12.

In some cases we have found that better results can be

obtained in making forecasts for a year ahead by using

L = 3 in place of 12. See Table 9 for the appropriate

comparisons. However, no clear cut rule can be recommended.

6 . Conclusions

For the forecasting problem at hand (see section 1)

,

if one can live with the percentage range of errors

indicated with the present NHTSA 's methods, there is no

coercive evidence to lead to abandoning the present

methods in favor of readily available computer routines

for exponential smoothing methods.

Of the methods examined in this study, the best results

were obtained with the EXPSMOOTHING routine using unadjusted

fatality data.

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APPENDIX A

Exponential Methods for Analysis and Forecasting

of Economic Time Series

From approximately 1960, most methods of time

series smoothing analysis have gone toward the use of

varying weights based on the age of the data. The simple

moving average ignores data beyond a certain age and

weights equally each datum that is used. The most popular

weighting schemes in vogue today are based on some form

of "exponential" smoothing. These schemes use all of the

data, with the weight given an individual datum decreasing

with increasing age of the datum. The rate of decreasing

weight is determined by a parameter called a smoothing

constant and most frequently denoted by the Greek letter

alpha (a). Depending on the nature of the time series,

a variety of models may be used to represent the time

series, for example, a constant process model = a^,

a linear process model = aQ + a-^t, a quadratic process

model C (t) = a^ + a^t + ^2^^' each case there

^n A is a process C (t) = l^_q a.j_t /±l which is observed in

the presence of noise x(t) = ? (t) + e (t) . The number

of "degrees of freedom" of these models are respectively

1, 2, 3, etc., and this number is sometimes referred

to as the order of the model . Multiple Smoothing ; If a constant model (C^ = a) is

used to represent the time series x(t) = ? (t) + t (t)

,

A-1
exponential smoothing for estimating the single coefficient

in a constant model is

a. = S. (x) - ax. + (1-a) . (x)

t t t t-1

For estimating the two coefficients! in a linear model,

the notion of double smoothing is used. For generality

the exponential smoothing for the constant model is

called single smoothing and is represented by

[1] [1]

S ^ (x) , i.e., S ^

(x) = S^(x). Then double smoothing

is defined as S^^^(x) = S^i^(x) + (1-a) S^^^ (x) , t t t-1

Similarly, multiple smoothing of order k is defined by

S^^^ (X) = (X) + (1-a) S (x),

t t t-1

th I.e., k— - order smoothing is just simple exponential

smoothing applied to the results of (k-l)st - order

smoothing of the data.

As an example, in the case of a linear model:

= a^ + a^t

the forecast of the value of the time series t units

after time T is given by

x^ (T) = a^ (T) + xa-^ (T)

A. /N where (T) and a^(T) are given by

a (T) = 2 S (x) - S (x)

o T T

ai(T) - a [s* (X) - S^f^^ (X)

1-a -"^

A2
A convenient, comprehensive basic reference is

R.G. Brown, Smoothing / Forecasting / and Prediction of

Discrete Time Series, Prentice-Hall, 196 3. A summary

of the fundamental formulas of multiple smoothing and

forecasting are given on pages 142-144, and pages 184ff

of this reference.

We remark at this time that in using multiple

smoothing methods two problems arise: (a) what order

of smoothing should be used? (b) what value of the

smoothing constant should be employed? A third, but

minor, problem is the determination of starting values.

The mathematical basis for the EXPSMOOTHING computer

program is founded on the concepts of multiple smoothing,

and utilizes the formulas presented in Brown's book.

A similar but different approach forms the basis

of the UNIVAC GEXSMO program. This generalized exponential

smoothing program is based on the work of P.P.. Winters,

"Forecasting Sales by Exponentially Weighted Moving

Averages," Management Science , April, 1960. The essentials

of this computer program are given in Appendix B under

the section UNIVAC 1108.

A3

APPENDIX B

Available Computer Programs at NBS for

Time Series Analysis

Several time-sharing interactive systems were

immediately available at NBS. These were DIALCOM,

Computer Sciences Corporation Conversational Executive

and General Electric Mark II. In addition the Large

Scale Systems STAT-PACK for the UNIVAC 1108 tests

several routines for time series analyses, A summary

of the pertinent program libraries of each of these

systems follows.

DIALCOM ; Moving Average (A Simple Moving average)

Autocovariance

Cross Covariance

Smoothed Series (A weighted moving average)

Seasonal Index and Cyclical Movement

Note: No expontential smoothing method is available and

none of the programs have an extrapolation feature.

Computer Sciences Conversational Executive (CSCX

Basic Library) ; This has a Triple Exponential Smoothing

routine called ***SMOOTH. In addition to smoothing, and

listing the smoothed values and differences, the routine

can produce a plot of the observed values and the smoothed

values (on the same graph) and an extrapolation feature

is also available.

B-1
There is no routine available for first or second

order smoothing and there is no automatic feature such

as a mean absolute deviation. The value of the smoothing

constant must be supplied by the user.

General Electric (Mark II) ; The EXPSMOOTHING

routine is a very comprehensive routine which performs

a sequence of exponential smoothing operations for a number of alpha values for each order of smoothing: 1,

2 1 3. The routine then selects that combination of

alpha and order of smoothing that has minimum mean

absolute deviation per data point. A forecast (extrapolation)

option is also available, along with a plot. A variety

of output options is possible.

The routine also permits the users to override

several automatic features of the routine. For example,

the user has the capability to specify cycle length

of known periodic data and also to specify the values

of the smoothing constants, but not the order of smoothing.

A summary of the essential details of this particular

routine is given in Appendix C.

UNIVAC 1108 : The Large Scale Systems STAT-PACK

for the UNIVAC 1108 lists several routines for time

series analysis. These are UNIVAC system subroutines

and are not necessarily National Bureau of Standards

routines, but they are available for use on tbo NBS

UNIVAC 1108. The most appropriate subroutine for the

problem at hand appears to be GEXSMO - Generalized

B-2
Exponential Smoothing.* The routine

1. Produces the seasonally adjusted series point

2. Computes a seasonality factor to be used

in the next cycle

3. Computes the next trend value

4. Produces a forecast for the next period,**

S

5. Compares the forecast with the actual to get

the forecast error

^t,l " ^t+1 " ^t,l

where S^_^^ is the (t+l)st time series data point.

When all the data are used, L forecasts for future

periods may be produced by using the equation

K

=

^^N + ^N-L+K ^ = 1' 2, L (Linear_ r increasing)

where L is the periodicity of the data, e.g., if the data

is by month, L = 12 and N is the number of elements in

the time series.

Three exponential smoothing factors, A, B, and C

all between 0 and 1 are used respectively to produce

the seasonally adjusted series, to adjust the seasonality

factors and to adjust the trend. These smoothing factors

and initial values may be input or may be calculated by

the routine.

*The notation used here is that employe.^ in the UNIVAC

Programmers Reference Manual. The mathematical background for

the method as given in Winters, P.R.: Forecasting Sales by

Exponentially Weighted Moving Averages, Management Science , April, 1960.

**In this context "next period" refers to the next time point.
The output of the GEXSMO routine is listed under

the headings:

ORIGINAL SEASONALLY FORECASTED SEASONALITY TREND ERROR

SERIES ADJ. SERIES SERIES

and this is followed by

MEAN ABSOLUTE DEVIATION OF FORECAST ERRORS

PERCENTAGE ERROR

We mention the availability of other UNIVAC time

series routines:

1. MOVAVG - Moving Averages. This subroutine

computes the smoothing coefficients in a polynomial

model for a time series and smooths the given

time series

.

2. SEASHI - Shiskin's Seasonality Factors. This

subroutine produces seasonality factors from

a time series, and smooths, detrends and

deseasonalizes the series.

3. WEMAV - Weighted Moving Averages, This subroutine

eliminates a trend from a time series by a

weighted moving average.

4. TRELS - Trend Analysis by Least Squares. This

subroutine removes trends in a time series with

a general linear model, using least squares.

B-4
5. VADIME - Variate Difference Method. This subroutine estimates the variance of the random

component in a time series and determines a lower

limit for the degree of the polynomial which

can be used in approximating the trend.

6. TSFARG - Autoregressive Model. This subroutine

obtains the least square coefficients in an

autoregressive time series model and produces

forward forecasts from the model.

The documentation of the detail of each of these

seven programs is very adequately done in the UNIVAC

Programmer's Inference Manual for STAT-PACK and is not

repeated here.

B-5

APPENDIX C

Details of EXPSMOOTHING Routine

The EXPSMOOTHING routine is a very comprehensive

routine which performs a sequence of exponential smoothing

operations for a number of alpha values for each order

of smoothing: 1, 2, 3. The routine then selects that

combination of alpha and order of smoothing that has

minimum mean absolute deviation per data point. A

forecast (extrapolation) option is also available, along

with a plot. A variety of output options is possible.

The routine also permits the users to override

several automatic features of the routine. For example,

the user has the capability to specify cycle length of

known periodic data and also to specify the values of

the smoothing constants, but not the order of smoothing.

For the convenience of the reader we present a summary of the essentials of the General Electric Time

Sharing EXPSMOOTHING Routine. The discussion is based

on experience in using the routine and explanations of

this routine from two User's Guides* prepared by the

*Mark II User's Guide: Statistical Analysis System,

General Electric Company, December, 1971 (5707.01).

GETSA$ : Mark I Marketing and Economic Forecasting,

A Hands-On Users Guide, General Electric Company, 1969.

(906329)

C-1
the General Electric Company, and is prepared in the

sequence provided by the version in use in December, 1972.

For additional details on program operation the reader

is referred to the user's guides. The time series data

to be subject to the EXPSMOOTHING routine is called RAW

DATA by the program. The RAW DATA may be adjusted

initially by what is called a BASE SERIES. This series

permits one to remove known distortions that one would

not expect to occur at future times. The base series

may be used to take into account discontinuities, human

judgment or the results of statistical analyses of the

data. The base series values are subtracted from the

raw data before an analysis is made and later added

back into the forecast.

To operate the program, in addition to entering

the raw data and the base series, one has to indicate

two times, the LEAD TIME (L) and the FORECAST HORIZON

(H) as well as a choice of six output options.

Lead time is the term used to describe the time

(in periods) between the last piece of data one has as

data as input and the period one is most interested in

estimating correctly. The forecast will be optimized

according to the lead time that is entered.

The forecast horizon indicates how many periods

beyond the last data point will be forecasted, but has

C-2
nothing to do with the optimization. For example, if

the raw data is reported monthly and the user wishes

to forecast six months in the future, accurately, and

also wishes to take a casual glance at the forecast one

year ahead, a lead time of six and a horizon time of 12

would be indicated.

The output options are cumulative, so that option

code 5 gives all options. The codes for the output

options are:

0 = Simple forecast table

1 = Raw data residue

2 = Raw data list

3 = Base series list and cyclic analysis

4 = Cyclic forecast

5 = Trend and error analysis and composite forecast.

A graphical display of the raw data and forecasted

(smoothed) and extrapolated data may also be obtained.

The following discussion is based on maximum output

(Code 5). The output indicates whether the smoothing

constant was supplied or provided automatically by the

programs and

a. NUMBER OF RAW DATA POINTS

b. NUMBER OF BASE DATA POINTS

c. FORECAST HORIZON

d. LEAD TIME.

Next, the RAW DATA are printed out.

C-3
Then comes DETERMINATION OF THE OPTIMUM NUMBER OF

DATA POINTS PER CYCLE. The entries are in columns K

and ERR(K), where K refers to cycle length as expressed

in data periods, and where K runs from 1 to half the

number of data points. The ERR(K) shows a measure*

of error associated with each cycle length. The GETSA$

user's guide states:

The lower the error, the stronger is the

tendency for the data to conform to a cyclic

pattern. The absolute value of the error

term has little significance for interpretation,

but the relative value shows which cycle

length fits best and that length (K value)

is chosen as the optimum length and subsequent

calculations assume a cycle of that length....

The optimum number of periods per cycle will

be chosen based on minimum error. That cycle

length is assumed for the remainder of the

analyses

.

If min ERR(K) occurs for K = 1, the routine will indicate

that the data is not cyclic.

Then the output indicates THE OPTIMUM PERIOD FOR

THE CYCLE and prints out the corrections that are used

to remove the cyclic effects, under the headings T and

C (T) **.

*We have been unable to determine the exact nature

of this measure.

**We have been unable to determine the exact nature

of this cdrrection.

C-4
Next there appears TABULATION OF THE ACTUAL

(UNSMOOTHED) RESIDUE. The residue at this stage contains

the corrections for base series and cyclic effects. At

this point the

Residue (T) =

T

Raw Data Residue (T) - I C{t-1),

where C(0) = 0.

The output then indicates the results of EXPONENTIAL

SMOOTHING APPLIED TO RESIDUE: under the headings

ALPHA TYPE OF SMOOTHING MEAN ABSOLUTE

DEVIATION PER DATA

POINT

If the alpha values are not supplied but are determined

by the program, the program will select* eight (8)

values of alpha and do exponential smoothings of orders

1, 2f and 3 for each alpha. The minimum absolute deviation

per data point for this set of 24 smoothings then

determines the alpha value and order of smoothing that

will be employed in the so-called FORECAST section of the

program. The output headings at this stage are:

OPTIMUM TYPE OF MEAN ABSOLUTE DEVIATION

ALPHA SMOOTHING PER DATA POINT

followed by FORECAST

TIME RESIDUE COMPOSITE ACTUAL ERROR

*If a is not forced, program will select 8 values of

a based on a = 1^ 2 . The leading term is always

8 L+F~

a-^ = .01. The second term is a2 = min (.Ob, a).

If a2 = .05, a-^ = OL r ~ 2°^' •••f o'g ~ 6a,

If a2 = a, a3 = 2a, a4 = 3^, ag = 7a.

C-5
Under TIME the initial entry is at time 5 + L, where L is

the lead time. Several observations are required before

a smoothed average can be calculated and this routine

uses five.

The RESIDUE indicated is the smoothed residue.

Under COMPOSITE appear the results of putting back

in the reverse corrections for cyclic effects and base

series.

Under ACTUAL are listed the values that were originally

listed as RAW DATA.

Under ERROR appear the results of ACTUAL minus

COMPOSITE. The entries under RESIDUE and COMPOSITE

beyond the last ACTUAL data point are based on the

exponential smoothing forecast formulas* of orders

1, 2, 3:

Fl^ = SI. = CEDl^

t+L t t

F2 t+L = CED2t + L(C2^)

F3^,^ = CED3. + L(C3^) + 1/2 l2(RC3.)

The values that are actually used in the extrapolation

are listed under the headings:

SI S2 S3

CEDl CED2 CED3

C2 C3 RC3

The values shown are for the last raw data point.

*These and other formulas used in the exponential smoothing,

process can readily be transformed into the notation used

in Brown's book in pages 142-144 and pages 184ff.

C-6
This is followed by HORIZON FORECAST BEGINS AT TIME

and LEAST SQUARES FIT

and MEAN and VARIANCE.

The LEAST SQUARES FIT is simply an equation describing

the straight line trend which best approximates (in the

sense of least squares) movement of the raw data for

all the data.

At this time the computer will ask WANT A PLOT?

and WANT DATA STORED? If a plot is desired, this routine

will give on one graph plots of the values listed under

FORECAST COMPOSITE and ACTUAL.

C-7

FORM HBS-114A 11.71) U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

S'.IEET

1. riiin.ICATIO.N OR REPORT NO.

NBGIR - 73-138

2. Gov't Accession 3. Recipient's Accession .\'o.

4. TITLE AND SUBTl I LE Time Series Forecasting of Highway Accident

Fatalities

5. Publication Date

6. r-^trforming Organization Code

7. AUTHOFaS)

Dr. Alexander R. Craw

8. Performing Orga-'ization

NBSIR - 73-13 3 | 9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BURFAU OF STANDARDS

DEPARTMENT OF COMMKRCE

WASHINGTON, D.C. 20234

10. Project/Task,' Work Unit \v.

4314496

11. Contract/Grant No.

12. Sponsoring Organization Name and Address

National Highway Traffic Safety Administration

Department of Transportation

Washington, D.C. 2 0590

13. Type of Report & Period Covered

14. Sponsoring Agency Code

15. SUPPLEMENTARY NOTES

16- ABSTRACT (A 200-word or less factual summary of most significarjt information. If document includes a significant

bibliography or literature survey, mention it here.) Using tv^elve years of time series data on highway fatalities,

Lhe methodology currently employed by the National Highway Traffic

3afety Administration (NHTSA) to forecast the annual (calendar

^ear) total of highway accident fatalities were compared with those

obtained by several computer routines based on exponential smoothing

techniques and available at the National Bureau of Standards. The

ase of unadjusted and seasonally adjusted data was also examined.

It is found that there is no coercive evidence to lead to

abandoning the present NHTSA methods in favor of readily available compute

routines based on exponential smoothing methods.

Of the methods examined in this study, the best results v/ere

obtained with the EXPSMOOTHING routine using unadjusted fatality data.

17. KEY WORDS (Alphabetical order, separated by semicolons)

Exponential smoothing; fatalities; highway accidents; time series

18. .AVAILABILITY STATEMENT

rXl UNLIMITED.

! I FOR OFFICIAL DISTRIBUTION. DO NOT RELEASE

TO NTIS.

19. SECURITY CLASS

(THIS REPORT)

UNCLASSIFIED

20. SECURIPr' CLASS

(THIS PAGE)

UNCLASSIFIED

21. NO. OF PAGES

22. Price

USCOMWDC SbZ'Ja.P"'!


III